On the Fine Interior of Three-dimensional Canonical Fano Polytopes

Batyrev, Victor; Kasprzyk, Alexander; Schaller, Karin

doi:10.48550/arxiv.1911.12048

Cited by 2 publications

(5 citation statements)

References 12 publications

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“…For example, the 4-dimensional almost reflexive lattice polytope P ⊂ R 4 : The suggested method for finding minimal models was tested for all 674 688 threedimensional Newton polytopes with one interior lattice point which were classified by Kasprzyk [Kas10]. The corresponding minimal surfaces Z are various smooth projective surfaces of non-negative Kodaira dimension κ with p g = 1: K3-surfaces (κ = 0), elliptic surfaces (κ = 1), Todorov and Kanev surfaces (κ = 2) [BKS19].…”

Section: P P *mentioning

confidence: 99%

“…Remark 2.10. There are 661 280 = 665 599 − 4 319 examples of three-dimensional lattice polytopes P with F (P ) = {0} which are not canonically closed, i.e., they are not reflexive [BKS19]. However, the canonical hulls C(P ) of these three-dimensional polytopes P are always reflexive.…”

Section: Canonically Closed Polytopesmentioning

confidence: 99%

“…Example 2.14. There are exactly 9 examples of 3-dimensional lattice polytopes P with F (P ) = ∅ and without interior lattice points [BKS19]. All these 9 polytopes are canonically closed, because their facets contain at least one lattice point in their relative interiors.…”

Section: Canonically Closed Polytopesmentioning

confidence: 99%

“…Therefore, it looks more reasonable to ask about some qualitative properties of the corresponding 3-dimensional minimal Calabi-Yau models such as possible types and numbers of isolated terminal cDV -singularities and possible non-integral values of their stringy Euler numbers [DR01]. It was shown in [BKS19] that there exist exactly 5 three-dimensional Newton polytopes defining Enriques surfaces. It is a reasonable task to extend this classification in dimension 4 and to obtain a complete list of all 4-dimensional lattice polytopes P with dim F (P ) = 0 and F (P ) ∩ M = ∅.…”

Section: The Stringy E-function Of a Minimal Modelmentioning

confidence: 99%

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Canonical models of toric hypersurfaces

Batyrev¹

2020

Preprint

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Let Z ⊂ T d be a non-degenerate hypersurface in d-dimensional torus T d ∼ = (C * ) d defined by a Laurent polynomial f with a given d-dimensional Newton polytope P . It follows from a theorem of Ishii that Z is birational to a smooth projective variety X of Kodaira dimension κ ≥ 0 if and only if the Fine interior F (P ) of P is nonempty. We define a unique projective model Z of Z having at worst canonical singularities which allows us to obtain minimal models Z of Z by crepant morphisms Z → Z. Moreover, we show that κ = min{d − 1, dim F (P )} and that general fibers in the Iitaka fibration of the canonical model Z are nondegenerate (d − 1 − κ)-dimensional toric hypersurfaces of Kodaira dimension 0. Using the rational polytope F (P ), we compute the stringy E-function of minimal models Z and obtain a combinatorial formula for their stringy Euler numbers.

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Section: P P *mentioning

confidence: 99%

Section: Canonically Closed Polytopesmentioning

confidence: 99%

Section: Canonically Closed Polytopesmentioning

confidence: 99%

Section: The Stringy E-function Of a Minimal Modelmentioning

confidence: 99%

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Canonical models of toric hypersurfaces

Batyrev¹

2020

Preprint

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“…All 3-dimensional canonical Fano polytopes are classified by Kasprzyk [Kas10]. Among them exist exactly 9089 canonical Fano polytopes ∆ such that dim F (∆) ≥ 1 [BKS19].…”

Section: E-polynomials Of Non-degenerate Hypersufacesmentioning

confidence: 99%

On the stringy Hodge numbers of mirrors of quasi-smooth Calabi-Yau hypersurfaces

Batyrev

2020

Preprint

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Mirrors X ∨ of quasi-smooth Calabi-Yau hypersurfaces X in weighted projective spaces P(w 0 , . . . , w d ) can be obtained as Calabi-Yau compactifications of non-degenerate affine toric hypersurfaces defined by Laurent polynomials whose Newton polytope is the lattice simplex spanned by d + 1 lattice vectors v i satisfying the relation i w i v i = 0. In this paper, we compute the stringy E-function of mirrors X ∨ and compare it with the Vafa's orbifold E-function of quasi-smooth Calabi-Yau hypersurfaces X. As a result, we prove the equalities of Hodge numbers h p,q str (X ∨ ) = h d−1−p,q orb (X) for all p, q and d as it is expected in mirror symmetry.

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On the Fine Interior of Three-dimensional Canonical Fano Polytopes

Cited by 2 publications

References 12 publications

Canonical models of toric hypersurfaces

Canonical models of toric hypersurfaces

On the stringy Hodge numbers of mirrors of quasi-smooth Calabi-Yau hypersurfaces

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